On hyperbolicity of SU(2)-equivariant, punctured disc bundles over the complex affine quadric

TL;DR

This paper studies the hyperbolic properties of SU(2)-equivariant punctured disc bundles over the complex affine quadric, identifying a maximal Stein bundle containing entire curves and showing others are Kobayashi hyperbolic.

Contribution

It classifies all SU(2)-equivariant punctured disc bundles over the complex affine quadric and determines their hyperbolic nature, highlighting the unique maximal bundle with entire curves.

Findings

The maximal SU(2)-equivariant Stein punctured disc bundle contains entire curves.

All other such punctured disc bundles are Kobayashi hyperbolic.

The classification provides insight into the hyperbolic geometry of these bundles.

Abstract

Given a holomorphic line bundle over the complex affine quadric $Q^2$, we investigate its Stein, SU(2)-equivariant disc bundles. Up to equivariant biholomorphism, these are all contained in a maximal one, say $\Omega_{max}$. By removing the zero section to $\Omega_{max}$ one obtains the unique Stein, SU(2)-equivariant, punctured disc bundle over $Q^2$ which contains entire curves. All other such punctured disc bundles are shown to be Kobayashi hyperbolic.